Satisfying more than half of a system of linear equations over GF(2): A multivariate approach

  • Authors:

  • R. Crowston;M. Fellows;G. Gutin;M. Jones;E. J. Kim;F. Rosamond;I. Z. Ruzsa;S. Thomassé;A. Yeo

  • Affiliations:

  • Royal Holloway, University of London, Egham, Surrey, TW20 0EX, UK;Charles Darwin University, Darwin, Northern Territory 0909, Australia;Royal Holloway, University of London, Egham, Surrey, TW20 0EX, UK;Royal Holloway, University of London, Egham, Surrey, TW20 0EX, UK;LAMSADE, CNRS UMR 7243, Université de Paris Dauphine, 75775 Paris, France;Charles Darwin University, Darwin, Northern Territory 0909, Australia;Rényi Institute of Mathematics, Hungarian Academy of Sciences, H-1053, Budapest, Hungary;LIP, UMR 5668, ENS de Lyon, CNRS, UCBL, INRIA, Université de Lyon, France;Singapore University of Technology and Design, Singapore

  • Venue:
  • Journal of Computer and System Sciences
  • Year:
  • 2014

Quantified Score

Hi-index 0.00

Visualization

Abstract

In the parameterized problem MaxLin2-AA[k], we are given a system with variables x"1,...,x"n consisting of equations of the form @?"i"@?"Ix"i=b, where x"i,b@?{-1,1} and I@?[n], each equation has a positive integral weight, and we are to decide whether it is possible to simultaneously satisfy equations of total weight at least W/2+k, where W is the total weight of all equations and k is the parameter (it is always possible for k=0). We show that MaxLin2-AA[k] has a kernel with at most O(k^2logk) variables and can be solved in time 2^O^(^k^l^o^g^k^)(nm)^O^(^1^). This solves an open problem of Mahajan et al. (2006). The problem Max-r-Lin2-AA[k,r] is the same as MaxLin2-AA[k] with two differences: each equation has at most r variables and r is the second parameter. We prove that Max-r-Lin2-AA[k,r] has a kernel with at most (2k-1)r variables.